295 \begin{axiom}[Product (identity) morphisms, preliminary version]

   296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, 

   297 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.

   298 These maps must satisfy the following conditions.

   299 \begin{enumerate}

   300 \item

   301 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram

   302 \[ \xymatrix{

   303 	X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\

   304 	X \ar[r]^{f} & X'

   305 } \]

   306 commutes, then we have 

   307 \[

   308 	\tilde{f}(a\times D) = f(a)\times D' .

   309 \]

   310 \item

   311 Product morphisms are compatible with gluing (composition) in both factors:

   312 \[

   313 	(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D

   314 \]

   315 and

   316 \[

   317 	(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .

   318 \]

   319 \item

   320 Product morphisms are associative:

   321 \[

   322 	(a\times D)\times D' = a\times (D\times D') .

   323 \]

   324 (Here we are implicitly using functoriality and the obvious homeomorphism

   325 $(X\times D)\times D' \to X\times(D\times D')$.)

   326 \item

   327 Product morphisms are compatible with restriction:

   328 \[

   329 	\res_{X\times E}(a\times D) = a\times E

   330 \]

   331 for $E\sub \bd D$ and $a\in \cC(X)$.

   332 \end{enumerate}

   333 \end{axiom}

   334 

   335 We will need to strengthen the above preliminary version of the axiom to allow

   336 for products which are ``pinched" in various ways along their boundary.

   337 (See Figure xxxx.)

   338 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs}.)

   339 Define a {\it pinched product} to be a map

   340 \[

   341 	\pi: E\to X

   342 \]

   343 such that $E$ is an $m$-ball, $X$ is a $k$-ball ($k<m$), and $\pi$ is locally modeled

   344 on a standard iterated degeneracy map

   345 \[

   346 	d: \Delta^m\to\Delta^k .

   347 \]

   348 In other words, \nn{each point has a neighborhood blah blah...}

   349 (We thank Kevin Costello for suggesting this approach.)

   350 

   351 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m{-}k$-ball,

   352 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension

   353 $l \le m-k$.

   354 

   355 It is easy to see that a composition of pinched products is again a pinched product.

   356 

   357 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction

   358 $\pi:E'\to \pi(E')$ is again a pinched product.

   359 A {union} of pinched products is a decomposition $E = \cup_i E_i$

   360 such that each $E_i\sub E$ is a sub pinched product.

   361 (See Figure xxxx.)

   362 

   363 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product

   364 $\pi:E\to X$.

   365 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories.

   366 In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$.

   367 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes in $X$, 

   368 taking inverse images under $\pi$ gives a cell decomposition of $E$ in which corresponding cells have the same codimension, and therefore accept the same labels.

   369 

   370 \nn{resume revising here; need to rewrite the following pasted axiom}

   371 

   372 \addtocounter{axiom}{-1}